 Hi, I'm Zor. Welcome to Unisor Education. We continue talking about limits, limits of the functions in this particular case. Sequences we have already covered. So these are limits of the functions. This lecture, as you know, is part of the course of Advanced Mathematics for teenagers and high school students. It's presented on Unisor.com. I do recommend you to watch this lecture from this website because it has notes, very detailed notes for each lecture, including this one. Also registered students can take exams, and this is a very good way to check how you basically master the material. The site is free, so. All right. Now, today I'm going to talk about two definitions which I have presented in the previous lecture. Two definitions of the concept of a function limit. Now, first of all, there are certain historical motivations for having these two separate definitions. Let me just remind you very briefly what they are. So if you have a function, let's have it graphically. This is my function. Now, this is some particular value, R, and we are talking about function converging to some value, A, as X converges to R. So let's just take, for instance, X1, X2, X3, etc. They are all converging to the R. And these are corresponding values of f at X1, f at X2, etc. And this is supposed to be A. So as X converges to R, function is converging to the value A. That is basically illustration of what the function limit actually is. And I have suggested two different definitions. One definition is the following. If for any sequence X1, X2, X3, etc., converging to R, any, any is a very, very important words right now. So if for any such sequence of arguments converging to R, corresponding any, this is the mathematical symbol for for any. If for any, the corresponding value of the function is a sequence which is converging to A, then A is a limit of the function f at X, at point, at point R. And this limit is equal to A. Another definition, now by the way, the good thing about this definition is that it's very I would say human-like. It's nature. I mean you see immediately that well, the X is moving towards R, f is supposed to move towards A and that seems to be corresponding to our intuitive understanding of what actually the limit is. And this important detail about any sequence which converges to R should really cause the corresponding sequence of the function values to converge to A. This is a very important thing because sometimes you can have functions and I give you an example that if you go by rational numbers, for instance, to the towards R function might take the value, let's say, 0, on irrational it takes the value 1. So it looks like there is no limit basically in this case. But nevertheless you have argument converging either by rational or by irrational numbers to the same R. So if and we cannot obviously say that in that particular case function has a limit, but if for any sequence converging to R of the arguments, my functions converge to A, then we can say that the function has a limit and its limit is equal to A. Another, oh yes, that's the positive thing. I forgot what negative. What's negative about this particular definition? Well, the negative is that how can I check that the function really has the limit based on this definition? Well, I have to go through all the different sequences of argument converging to R. But how can I check for all the sequences? It's impossible. There is an infinite number of sequences, obviously, which are converged to R. So I can't really constructively check the existence of the limit. I cannot prove that function has a limit because I cannot really exhaust all the sequences of the arguments which are converging to R. So there is another definition. It's not as human-like. It's not as natural as far as, you know, what we intuitively think about this. However, it's perfectly correct and it's equivalent to this one. And this is an equivalency, which I'm going to prove today. So let me remind you the second one. Now the second one is if I would like to be as close on my function as possible to the value a. So for any degree of closeness, epsilon, positive. However small, there should be, there should exist. So for any epsilon should exist delta, such that as soon as my argument is within delta from limit point, immediately following from this, my function would be close to the limit than this particular value epsilon. This is not as obvious as this one. However, it's also correct. And let me just try to explain it a little bit more over and beyond whatever I did in the previous lecture. Now this signifies the closeness of the argument to the R. Any kind of argument as long as it's within delta neighborhood as we are saying. So within very, very small. So for any delta, for any epsilon neighborhood of a, so if I would like my function to be from a minus epsilon to a plus epsilon somewhere here. There is always such a range of argument where this is true. And then I can make this smaller by vertical. So epsilon gets smaller. And then for a smaller, I still have to find an delta neighborhood of my R where all the values of the functions will be within this smaller neighborhood. So no matter how small a neighborhood I do, I would like to really be, I would like to be as close to a as possible, measuring is, measuring it by epsilon. For any measure, I will find certain degree of closeness to the R. So whenever my argument is very close to the R, my function will be within this narrow interval around A. So this is my other definition. And today I'm going to prove that these definitions are equivalent, which means if this function has limit A as X goes to R, according to one definition, then it's actually the same limit with the same condition according to another definition. And vice versa. So that's what I'm going to prove right now. Okay? All right. Now before starting this proof, I would like to mention something very important. I would like to, I would like to recommend you to try to prove it yourself before actually listening to my explanation. Whether you will be successful or unsuccessful doesn't really matter as long as you spend some time thinking about how to prove from one to another or from the other to the first one. It's very, it's very useful exercise for you. And you know, the whole purpose of this course is to basically convince you that this kind of a mathematics is really necessary for for your creativity, for your development of your logical thinking, your analytics, etc. It's not the purpose of just giving a certain amount of information. Nobody needs the proof. I mean, everybody knows that these are two equivalent definitions. So why do we prove it? Well, it's an excellent school of thought for you. So I would rather recommend you to stop this video if you didn't do it before. Stop this video. Go to Explanationonunisor.com of these two theorems. And there are proofs that as well. Don't read the proofs. Try to prove it yourself. Then again, after you spend at least an hour, that's what I would suggest. You can read whatever is written there, then you can listen to me. And when everything is finished, I do recommend you to try again. Just do it by yourself from the memory, whatever you have remembered from my explanation, or from reading the the website. Okay, now I will go. So the first one is, I would like to prove that if my, so my first definition is the following. If my Xn converges to R, and for any such sequence of Xn, if this is any sequence which is converging to R, if from this case, follows that my functions of these arguments represent a sequence which is converging to A, then the limit of the function is A. So let's consider that this function has limit A, which means that this is true. What I would like to prove is that for any epsilon greater than zero, there is such a delta that as long as my X is in delta neighborhood from R, my function would be within epsilon neighborhood of A. So this is given, this is a true statement, and this I have to prove. By the way, sometimes people are saying strictly less here and here. It doesn't really matter for definition. It doesn't really, both are completely equivalent. Alright, so how can I prove that? Alright. Let's assume that this is not true. What does it mean that this is not true? Well, it means that for some particular epsilon, there is no such delta. Right? Now, what does it mean that there is no such delta? It means that whatever value of delta I can choose, doesn't really matter which one. From this, this does not follow. So I chose some particular, so I'm negating this. So from some particular value of epsilon, there is no such delta that this is true. What does it mean that there is no such delta? It means that no matter what kind of delta I choose, if my X would be closer than this delta, my function would not be within this particular epsilon neighborhood. My function would be outside neighborhood. So I choose one particular epsilon. I put it as epsilon zero, it doesn't really matter. So there is no such delta that this is true. Which means that no matter what delta I choose, if this is true, this is false. Alright? Okay, fine. So let's choose delta equals to one, delta first equals to one. And I choose one particular X which is closer than one to R. I find this X1. And for this particular X1, I know that f of X minus A greater than epsilon. Now, I will choose another delta. One half. And choose X2 within one half of the R. Still, now this is f of X1. This is f of X2 also would be greater than epsilon. Delta three, I will choose one third. And X3 is within one third from the R. And I know that my f of X3 is still greater than epsilon, etc. So my delta nth is equal to one nth. I choose Xn which is within one nth from R. And then still my f of Xn. So what do I have right now? I have a sequence of X1, X2, Xn, etc. Which is converging to R. Which is this. It's true, right? Since my difference is less than one, less than two, one half, less than one third, less than one nth. So obviously the difference between Xn and R is infinitesimal. My function is still outside of the epsilon neighborhood around A. So I have actually constructed a sequence of arguments which are going to R. For which values of the functions do not converge to A. Because the difference between them and A is still greater than something. So they do not converge. So it's contradiction with my premise that any sequence, you see, this is what's important about this any. I said that any sequence of the argument which goes to R results in the sequence of arguments of the function which is converging to A. And now I'm explicitly constructing one particular sequence that this is not true. What does it mean? That my initial negation of this leads to the contradiction with my premise. So my initial negation is wrong. Which means that this is true. And that's the proof, basically. That's the end of the proof. There is no such epsilon for which something like this is happening. So for all epsilon, as long as my, there is always some kind of a delta where closeness to the argument to the limit point results in the closeness of the limit point. That's the function. All right? Well, that's one thing. That's the proof from the first definition to the second. Now let's try to do it the other way around. So this is given and this what we have to prove. Right? Okay, fine. We have to prove this for any sequence, right? Sequence of arguments. Well, let's take the sequence of arguments. Any sequence of arguments. Any. I just took any sequence of arguments which is converging to R. Now what happens then? Now I know that my, as long as my sequence converges to R, there is, for any delta, there is always some number N. After which, as long as N greater than N, my X N minus R greater than, less than or equal to delta. Right? That's what it means. Convergence to R means that whatever the degree of closeness I choose after certain number of N, all my members of my sequence will be within the delta neighborhood from R. That's what it means, right? That's basically, that's what given when I'm saying let's take any particular sequence which goes to R. Okay? Now, what I do have to prove is that the corresponding sequence of functions of these X N, that's what I have to prove. Now, what does it mean that I have to prove it? I have to prove that for any epsilon, there is always some number N such that as long as N greater than N or equal from this follows my function minus A would be less than epsilon, right? So, let me choose one particular, one particular value of delta. Now, I have this. Now, for any epsilon, for any epsilon, so I can choose this one. I can find such a delta that this is true, right? Because this is my premise of this particular theory. So, from this epsilon, I can always find delta. Now, from the delta, I can find number N when this is true. So, what happens? It means that no matter what kind of epsilon I choose, no matter what kind of epsilon I choose, based on this, I can find delta. Because that's what basically this definition states. That if the function has limit of A, it means that for any epsilon, there is a delta such that as long as your argument is within delta neighborhood, my function will be within epsilon neighborhood. So, for any epsilon, I can find my delta. Now, from delta, I can find number N because I know that this is converging to R. So, for all these numbers N, my X would be within this neighborhood. So, from this follows this, which means this would be within the epsilon neighborhood. And that's basically it. So, let me just try to maybe say it slightly differently. Because, again, what do I have to prove? I have to prove that for any epsilon, there is some kind of a number when this function would be in the epsilon neighborhood from the A. But for any epsilon, including whatever I choose, I can always find delta. And based on this definition, I can find the delta neighborhood of R, where my argument should be to necessitate this. And since my X is converging to R, I can always, by this delta, I can always find N. After which, all my members, including these ones, would be within delta and this would be within epsilon, therefore. So, that's my proof from this more constructive way, the one which is, I would say, more intuitive but less constructive. Okay, I hope I explained it in a relatively understandable terms. If still it's not exactly what you would prefer, go to thisunisor.com and read the explanation as it is written. The proofs are written over there. If you still do not understand it, send me an email and I'll try to personally explain maybe in writing again something which is not really clear. But in any case, it's very important for you to really think about both proofs because it's excellent exercise in logic. Because here you have some logical statement. For the first proof, I had to negate that statement. And there are a little bit more complicated maybe than you're used to logical considerations here. So, that's why I would like you really to spend some time on this. It's much more useful to spend time thinking about this than to memorize formulas or whatever else. It's very important to understand completely how these both logical conclusions can be derived one from another and backwards. So, these two definitions are equivalent. If you feel comfortable about these proofs, I guarantee that all other things related to derivatives will probably go very, very smoothly for you. Alright, that's it. Thanks very much and good luck.